Douglas W. Oldenburg

3-D inversion of magnetic data

We present a method for inverting surface magnetic data to recover 3-D susceptibility models. To allow the maximum flexibility for the model to represent geologically realistic structures, we discretize the 3-D model region into a set of rectangular cells, each having a constant susceptibility. The number of cells is generally far greater than the number of the data available, and thus we solve an underdetermined problem. Solutions are obtained by minimizing a global objective function composed of the model objective function and data misfit. The algorithm can incorporate a priori information into the model objective function by using one or more appropriate weighting functions. The model for inversion can be either susceptibility or its logarithm. If susceptibility is chosen, a positivity constraint is imposed to reduce the nonuniqueness and to maintain physical realizability. Our algorithm assumes that there is no remanent magnetization and that the magnetic data are produced by induced magnetization only. All minimizations are carried out with a subspace approach where only a small number of search vectors is used at each iteration. This obviates the need to solve a large system of equations directly, and hence earth models with many cells can be solved on a deskside workstation. The algorithm is tested on synthetic examples and on a field data set.

3-D inversion of gravity data

We present two methods for inverting surface gravity data to recover a 3-D distribution of density contrast. In the first method, we transform the gravity data into pseudomagnetic data via Poissons relation and carry out the inversion using a 3-D magnetic inversion algorithm. In the second, we invert the gravity data directly to recover a minimum structure model. In both approaches, the earth is modeled by using a large number of rectangular cells of constant density, and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data. The model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints. We apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information. Applications of the algorithms to synthetic and field data produce density models representative of true structures. Our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information.

The inversion and interpretation of gravity anomalies

A rearrangement of the formula used for the rapid calculation of the gravitational anomaly caused by a two‐dimensional uneven layer of material (Parker, 1972) leads to an iterative procedure for calculating the shape of the perturbing body given the anomaly. The method readily handles large numbers of model points, and it is found empirically that convergence of the iteration can be assured by application of a low‐pass filter. The nonuniqueness of the inversion can be characterized by two free parameters: the assumed density contrast between the two media, and the level at which the inverted topography is calculated. Additional geophysical knowledge is required to reduce this ambiguity. The inversion of a gravity profile perpendicular to a continental margin to find the location of the Moho is offered as a practical example of this method.

Estimating depth of investigation in DC resistivity and IP surveys

In this paper, the term “depth of investigation” refers generically to the depth below which surface data are insensitive to the value of the physical property of the earth. Estimates of this depth for dc resistivity and induced polarization (IP) surveys are essential when interpreting models obtained from any inversion because structure beneath that depth should not be interpreted geologically. We advocate carrying out a limited exploration of model space to generate a few models that have minimum structure and that differ substantially from the final model used for interpretation. Visual assessment of these models often provides answers about existence of deeper structures. Differences between the models can be quantified into a depth of investigation (DOI) index that can be displayed with the model used for interpretation. An explicit algorithm for evaluating the DOI is presented. The DOI curves are somewhat dependent upon the parameters used to generate the different models, but the results are robust...

Inversion of induced polarization data

We present an algorithm for inverting induced polarization (IP) data acquired in a 3-D environment. The algorithm is based upon the linearized equation for the IP response, and the inverse problem is solved by minimizing an objective function of the chargeability model subject to data and bound constraints. The minimization is carried out using an interior-point method in which the bounds are incorporated by using a logarithmic barrier and the solution of the linear equations is accelerated using wavelet transforms. Inversion of IP data requires knowledge of the background conductivity. We study the effect of different approximations to the background conductivity by comparing IP inversions performed using different conductivity models, including a uniform half-space and conductivities recovered from one-pass 3-D inversions, composite 2-D inversions, limited AIM updates, and full 3-D nonlinear inversions of the dc resistivity data. We demonstrate that, when the background conductivity is simple, reasonable IP results are obtainable without using the best conductivity estimate derived from full 3-D inversion of the dc resistivity data. As a final area of investigation, we study the joint use of surface and borehole data to improve the resolution of the recovered chargeability models. We demonstrate that the joint inversion of surface and crosshole data produces chargeability models superior to those obtained from inversions of individual data sets.

Recovery of the acoustic impedance from reflection seismograms

This paper examines the problem of recovering the acoustic impedance from a band‐limited normal incidence reflection seismogram. The convolutional model for the seismogram is adopted at the outset, and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible. In the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace (that is, the band‐limited version of the reflectivity function) into the standard recursion formula for the acoustic impedance. The formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance. Moreover, the averaging function is identical to that used in deconvolving the initial seismogram. The advantage of these averages is that they are unique; their disadvantage is that low‐frequency information, which is crucial to making a g...

JOINT INVERSION : A STRUCTURAL APPROACH

We develop a methodology to invert two different data sets with the assumption that the underlying models have a common structure. Structure is defined in terms of absolute value of curvature of the model and two models are said to have common structure if the changes occur at the same physical locations. The joint inversion is solved by defining an objective function which quantifies the difference in structure between two models, and then minimizing this objective function subject to satisfying the data constraints. The problem is nonlinear and is solved iteratively using Krylov space techniques. Testing the algorithm on synthetic data sets shows that the joint inversion is superior to individual inversions. In an application to field data we show that the data sets are consistent with models that are quite similar.

Simultaneous 1D inversion of loop–loop electromagnetic data for magnetic susceptibility and electrical conductivity

Magnetic susceptibility affects electromagnetic (EM) loop–loop observations in ways that cannot be replicated by conductive, nonsusceptible earth models. The most distinctive effects are negative in‐phase values at low frequencies. Inverting data contaminated by susceptibility effects for conductivity alone can give misleading models: the observations strongly influenced by susceptibility will be underfit, and those less strongly influenced will be overfit to compensate, leading to artifacts in the model. Simultaneous inversion for both conductivity and susceptibility enables reliable conductivity models to be constructed and can give useful information about the distribution of susceptibility in the earth. Such information complements that obtained from the inversion of static magnetic data because EM measurements are insensitive to remanent magnetization.We present an algorithm that simultaneously inverts susceptibility‐affected data for 1D conductivity and susceptibility models. The solution is obtaine...

A Discrimination Algorithm for UXO Using Time Domain Electromagnetics

An assumption is made that the Time Domain Electromagnetic (TEM) response of a buried axisymmetric metallic object can be modelled as the sum of two dipoles centered at the midpoint of the body. The strength of the dipoles depends upon the relative orientation between the object and the source field, and also upon the shape and physical properties of the body. Upon termination of the source field, each dipole is assumed to decay as k(t+α)−βe−t∕γ. The parameters k, α, β and γ depend upon the conductivity, permeability, size and shape of the object, and these can be extracted from the measurements by using a nonlinear parametric inversion algorithm. Investigations carried out using an analytic solution for a sphere and laboratory measurements of steel and aluminum rectangular prisms, suggest the following two-step methodology: (1) The value of β is first used as a diagnostic to assess whether the metallic object is non-magnetic or magnetic, (2) the ratios of k1∕k2 and β1∕β2 are then diagnostic indicators as...

Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach

We present a general formulation for inverting frequencyor time-domain electromagnetic data using an all-at-once approach. In this methodology, the forward modeling equations are incorporated as constraints and, thus, we need to solve a constrained optimization problem where the parameters are the electromagnetic fields, the conductivity model, and a set of Lagrange multipliers. This leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought. Nevertheless, experience shows that the constrained problem can be solved faster than the unconstrained one. The primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process, and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly. In this paper, we outline the all-atonce approach and apply it to electromagnetic problems in both frequency and time domains. This is facilitated by a unified representation for forward modeling for these two types of data. The optimization problem is solved by finding a stationary point of the Lagrangian. Numerically, this leads to a nonlinear system that is solved iteratively using a Gauss-Newton strategy. At each iteration, a large, indefinite matrix is inverted, and we discuss how this can be accomplished. As a test, we invert frequency-domain synthetic data from a grounded electrode system that emulates a field CSAMT survey. For the time domain, we invert borehole data obtained from a current loop on the surface.